Hansen–Vukićević conjecture on Zagreb indices

Let $\mathcal{G}$ be a simple finite graph with vertex set $V(\mathcal{G})$ and edge set $e(\mathcal{G})$. Its first and second Zagreb indices are

$$M_{1}(\mathcal{G})=\sum_{v\in V(\mathcal{G})}\deg(v)^{2},\qquad M_{2}(\mathcal{G})=\sum_{uv\in e(\mathcal{G})}\deg(u)\deg(v).$$

Hansen–Vukićević conjecture. For any simple finite graph $\mathcal{G}$,

$$\dfrac{M_{2}(\mathcal{G})}{\lvert e(\mathcal{G})\rvert}\geq\dfrac{M_{1}(\mathcal{G})}{\lvert V(\mathcal{G})\rvert}.$$

Hansen and Vukićević posed this inequality as a comparison between the first and second Zagreb indices. It was disproved by the example $\Gamma=K_{1,5}\sqcup K_{3}$, so the conjecture is refuted.